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A cone is a convex cone if + belongs to , for any positive scalars , , and any , in . [5] [6] A cone is convex if and only if +.This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algebraic, or (more commonly) the real numbers.
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec = ...
Cone of a circle. The original space X is in blue, and the collapsed end point v is in green.. In topology, especially algebraic topology, the cone of a topological space is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.
The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general Noetherian schemes. Let X be an algebraic variety, x a point of X, and (O X,x, m) be the local ring of X at x. Then the tangent cone to X at x is the spectrum of the associated graded ring of O X,x with respect to the m-adic filtration:
In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.
The vector spaces () and () are dual to each other by the intersection pairing, and the nef cone is (by definition) the dual cone of the cone of curves. [ 6 ] A significant problem in algebraic geometry is to analyze which line bundles are ample , since that amounts to describing the different ways a variety can be embedded into projective space.
Cone (graph theory), a graph in which one vertex is adjacent to all others; Cone (linear algebra), a subset of vector space; Mapping cone (homological algebra) Cone (topology) Projective cone, the union of all lines that intersect a projective subspace and an arbitrary subset of some other disjoint subspace
A more involved example is the role played by the cone of curves in the theory of minimal models of algebraic varieties. Briefly, the goal of that theory is as follows: given a (mildly singular) projective variety X {\displaystyle X} , find a (mildly singular) variety X ′ {\displaystyle X'} which is birational to X {\displaystyle X} , and ...